Traditionally, the numerical performance of the lattice Boltzmann method is mainly determined by the moment degree of a discrete equilibrium distribution. The equilibrium distribution positivity is merely considered as an ancillary property which is used to constrict the numerical configuration. With the newly-developed partial Gaussian-Hermite quadrature scheme, the positivity of equilibrium distribution is validated as an independent property like moment degree which can be adjusted by discrete velocities. Researchers speculated that the positivity should also be significant for the numerical performance by the lattice Boltzmann method and can be used to improve the performance. Comparing with the classical improvement through moment degree, the positivity approach will not bring additional computation. However, due to the lack of boundary treatment, the speculation has not been validated by detailed numerical simulations. In this paper, through employing a periodic case, the Taylor-Green vortex, to avoid the boundary issue, we in depth analyze the numerical effect of the model positivity, including the numerical accuracy in the model positive range, the influence of positivity on the numerical performance, and the significance comparison between positivity and moment degree. The results show that for a given model, the numerical accuracy is not consistent in the whole positive range. As the configuration is close to the border of positive range, the accuracy will degrade though it is still acceptable. The numerical performance of a model depends on both moment degree and positivity. The role that the moment degree plays lies mainly in the qualification of a model on Galilean invariance. Once a model fulfills the Galilean invariance, its numerical performance is solely dependent on the positivity. Hence, the improvement approach through modifying the model positivity is a viable solution, and a Galilean invariant model with wider positive range does possess a better numerical performance regardless of its moment degree. Furthermore, based on the numerical results in this paper, all DnHm models with high moment degree are better than the classical D2Q9 model. Of the above models, the D2H3-2 model has the best performance and deserves to be further studied